Self-similar prior and wavelet bases for hidden incompressible turbulent motion

Patrick Héas 1 Frédéric Lavancier 2 Souleymane Kadri-Harouna 3
1 ASPI - Applications of interacting particle systems to statistics
IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique
2 Probabilités, statistique et calcul scientifique
LMJL - Laboratoire de Mathématiques Jean Leray
Abstract : This work is concerned with the ill-posed inverse problem of estimating turbulent flows from the observation of an image sequence. From a Bayesian perspective, a divergence-free isotropic fractional Brownian motion (fBm) is chosen as a prior model for instantaneous turbulent velocity fields. This self-similar prior characterizes accurately second-order statistics of velocity fields in incompressible isotropic turbulence. Nevertheless, the associated maximum a posteriori involves a fractional Laplacian operator which is delicate to implement in practice. To deal with this issue, we propose to decompose the divergent-free fBm on well-chosen wavelet bases. As a first alternative, we propose to design wavelets as whitening filters. We show that these filters are fractional Laplacian wavelets composed with the Leray projector. As a second alternative, we use a divergence-free wavelet basis, which takes implicitly into account the incompressibility constraint arising from physics. Although the latter decomposition involves correlated wavelet coefficients, we are able to handle this dependence in practice. Based on these two wavelet decompositions, we finally provide effective and efficient algorithms to approach the maximum a posteriori. An intensive numerical evaluation proves the relevance of the proposed wavelet-based self-similar priors.
Type de document :
Article dans une revue
SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2014, 7 (2), pp.1171-1209
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  • HAL Id : hal-00793461, version 2

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Patrick Héas, Frédéric Lavancier, Souleymane Kadri-Harouna. Self-similar prior and wavelet bases for hidden incompressible turbulent motion. SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2014, 7 (2), pp.1171-1209. 〈hal-00793461v2〉

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